WASHINGTON -- As House GOP Whip Kevin McCarthy (R-Calif.) quickly shored up support among his colleagues to succeed Eric Cantor (R-Va.) as majority leader, conservatives complained their members were at an unfair disadvantage due to snap elections.

House Republicans will vote June 19 on a replacement for Cantor, who will step down as majority leader on July 31 and serve out the remainder of the congressional session.

Cantor was defeated Tuesday by economics professor Dave Brat in the political shocker of the season.

He threw his weight solidly behind the effort of the No. 3 Republican in the House, McCarthy, to become No. 2.

"I can tell you that if my dear friend and colleague Kevin McCarthy does decide to run, I think he'd make an outstanding majority leader. And I will be backing him with my full support," Cantor told media on Wednesday. Indeed, McCarthy was huddling with aides and reaching out to House chairmen as soon as Cantor lost Tuesday night.

The race for majority leader has narrowed since then with Reps. Jeb Hensarling (R-Texas) and Pete Sessions (R-Texas) both squashing speculation that they would aim for the job.

"After prayerful reflection, I have come to the conclusion that this is not the right office at the right time for me and my family," Hensarling said in a statement, adding that he was "humbled" by the encouragement he received to run for Cantor's job.

House Republican Conference Chairwoman Cathy McMorris Rodgers (R-Wash.), currently No. 4 in the leadership, also has declined the chance to move up. She faces an August primary challenge from a political novice who plans to use the same illegal immigration platform that Brat used against Cantor.

Rep. Steve King (R-Iowa) said that calling a leadership election so soon "has the effect of stacking the deck."

“At a time when Republicans in Congress are fighting the Obama administration to oppose snap elections for unions on American employers, Republican leadership is trying to do the same in the United States House of Representatives,” said King.

"Unfortunately, while both current candidates benefit from the hasty timeframe prescribed by leadership, neither opposes amnesty legislation being brought to the floor of the House," he said Thursday morning when Sessions was still in the running. "The primary election in Virginia 7 that led to this leadership vacuum turned on the electorate’s opposition to amnesty. Have we learned nothing?"

"Let’s take more time to get our heads clear and elect a staunch conservative, anti-amnesty candidate to step up and lead the Majority.”

With the expectation that McCarthy will ascend and the whip position will be open, Rep. Pete Roskam (R-Ill.) has stepped up his longtime lobbying of colleagues for the No. 3 post. Rep. Marlin Stutzman (R-Ind.), who got the boot from the whip team for defying leadership on a procedural farm bill vote, is also vying for whip.

Republican Study Committee Chairman Steve Scalise (R-La.), who leads about 180 caucus conservatives, has not officially announced a bid but has been whipping up support among the right for a run at McCarthy's job.